Problem: Simplify; express your answer in exponential form. Assume $q\neq 0, r\neq 0$. $\dfrac{{(q^{-5}r^{2})^{3}}}{{(q^{-2}r^{-3})^{5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{-5}r^{2})^{3} = (q^{-5})^{3}(r^{2})^{3}}$ On the left, we have ${q^{-5}}$ to the exponent ${3}$ . Now ${-5 \times 3 = -15}$ , so ${(q^{-5})^{3} = q^{-15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{-5}r^{2})^{3}}}{{(q^{-2}r^{-3})^{5}}} = \dfrac{{q^{-15}r^{6}}}{{q^{-10}r^{-15}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{-15}r^{6}}}{{q^{-10}r^{-15}}} = \dfrac{{q^{-15}}}{{q^{-10}}} \cdot \dfrac{{r^{6}}}{{r^{-15}}} = q^{{-15} - {(-10)}} \cdot r^{{6} - {(-15)}} = q^{-5}r^{21}$